This paper sets up a language to deal with
Dirac operators on manifolds with corners of arbitrary codimension. In
particular the author develops a precise theory of boundary reductions.

The author introduces the notion of a taming of a Dirac operator as
an invertible perturbation by a smoothing operator. Given a Dirac
operator on a manifold with boundary faces the author uses the tamings
of its boundary reductions in order to turn the operator into a
Fredholm operator. Its index is an obstruction against extending the
taming from the boundary to the interior. In this way he develops an
inductive procedure to associate Fredholm operators to Dirac operators
on manifolds with corners and develops the associated obstruction
theory.